As feature sizes of semiconductor devices (e.g., very-large-scale integration (VLSI) circuits) become smaller, variations in these semiconductor devices become larger. Such variations include random variations and systematic variations. For example, random (e.g., uncorrelated) variations affect relative characteristics of closely-placed identical devices (often called matching or mismatch). Other device and circuit examples involving random variations include mismatch among multiple identical n-type field-effect transistors (NFETs) used in a ring oscillator, mismatch among multiple identical p-type field-effect transistors (PFETs) used in a ring oscillator, and mismatch among multiple identical fingers in a multi-finger radio frequency (RF) field-effect transistor (FET) or metal-oxide-semiconductor (MOS) varactor.
From the viewpoint of circuit modeling, systematic variations affect all instances of a device (e.g., all gate oxide thicknesses of NFETs) together, and there is no relative difference between any two instances of the device. On the other hand, random variations affect each instance of a device from any other instances of the device, e.g., FET threshold voltage (Vt) variation caused by random doping fluctuations, a random part of FET channel length variation, and mismatch among a group of nearby and identical resistors. In these cases of random variations, a relative difference between any two instances of a device is the subject of mismatch modeling.
For analog circuits, device mismatch is one of limiting factors for circuit performance. For semiconductor devices, there exist statistical models for device mismatch. For example, circuit designers can use a Monte Carlo model for device mismatch to simulate and evaluate an effect of device mismatch on circuit performance. More specifically, an electric performance target F in each of a group of N semiconductor devices (or circuits) is affected by one or more statistical parameters P, Q, R, S, . . . . Namely, a performance target F1 of a first device depends on statistical parameters P1, Q1, R1, S1, . . . , a performance target F2 of a second device is affected by statistical parameters P2, Q2, R2, S2, . . . , and a performance target FN of an Nth device is a function of statistical parameters PN, QN, RN, SN, . . . .Fi=f(Pi,Qi,Ri,Si, . . . ), i=1,2,3, . . . , N.  (1.0)
For each device (or circuit), one of statistical parameters (say, the parameter P) has a mismatch component, and other statistical parameters have global components only. For example, the electrical parameter P can be a threshold voltage of a metal-oxide-semiconductor field-effect transistor (MOSFET), a channel length of the MOSFET, a sheet resistance of one of a poly and a diffused resistor, or an area density of a capacitance of a capacitor, etc.
Mismatch among N performance targets F1, F2, . . . , FN are affected by mismatch among N statistical mismatch parameters P1, P2, . . . , PN. To model mismatch among the N parameters P1, P2, . . . , and PN, means and variances of the N parameters P1, P2, . . . , and PN and mismatch values among them may have several requirements. First, each of the devices may have to have a same predetermined mean μ for the parameter Pi, as determined by the following equation:Pi=μ, i=1,2,3, . . . , N.  (1.1)
Second, each of the devices may have to have a same variance for the parameter Pi, as determined by the following equation:
                                          〈                                          (                                                      P                    i                                    -                                      〈                                          P                      i                                        〉                                                  )                            2                        〉                    -                      σ            cm            2                    +                                    1              2                        ⁢                          σ                              m                ⁢                                                                  ⁢                0                            2                                      ,                  i          =          1                ,        2        ,        3        ,        …        ⁢                                  ,        N        ,                            (        1.2        )            where σcm is a standard deviation for chip mean (systematic, global, or correlated) part of variations (e.g., standard deviations) and σm0 is a standard deviation for mismatch part of variations.
Third, each pair of the devices may have to have a same mismatch value for the parameters Pi and Pj, as determined by the following equation:
                                          〈                                          (                                                      P                    i                                    -                                      P                    j                                                  )                            2                        〉                    =                                                    1                L                            ⁢                                                ∑                                      l                    =                    1                                    L                                ⁢                                                      (                                                                  P                                                  i                          ,                          l                                                                    -                                              P                                                  j                          ,                          l                                                                                      )                                    2                                                      =                          σ                              m                ⁢                                                                  ⁢                0                            2                                      ,                                  ⁢        i        ,                  j          =          1                ,        2        ,        3        ,        …        ⁢                                  ,        N        ,                  i          ≠          j                ,                            (        1.3        )            where L is a number of circuit simulation runs.
The Monte Carlo model for adjacent device mismatch may be determined by determining the parameter Pi for each of the devices, as follows:
                                          P            i                    =                      μ            +                                          σ                cm                            ⁢              G                        +                                          1                                  2                                            ⁢                              σ                                  m                  ⁢                                                                          ⁢                  0                                            ⁢                              g                i                                                    ,                  i          =          1                ,        2        ,        3        ,        …        ⁢                                  ,        N        ,                            (        1.4        )            where each of G, g1, g2, g3, . . . , gN is an independent stochastic or random variable of a mean of zero and a standard deviation of 1, σcmG represents a global component of statistical variation in Pi, and σm0gi represents a mismatch component of statistical variation in Pi.
The Monte Carlo model (1.4) satisfies the requirements (1.1), (1.2), and (1.3), and thus, can be a model for mismatch among the N statistical parameters P1, P2, . . . , and PN. Monte Carlo simulations are run using the Monte Carlo model to evaluate the effect of device mismatch on circuit performance. However, the Monte Carlo simulations take a long time to complete since they require up to tens of thousands of runs using a random number generator to determine the Gaussian distributions in the Monte Carlo model. In addition, most of the determined parameters and mismatches are not of interest because they are not worst-case values.
Accordingly, there exists a need in the art to overcome the deficiencies and limitations described hereinabove.